1. technology and computing

Summable series and the Riemann rearrangement
theorem
Jordan Bell
[email protected]
Department of Mathematics, University of Toronto
May 4, 2015
1
Introduction
Let N be the set of positive integers. A function from N to a set is called a
sequence. If X is a topological space and x ∈ X, a sequence a : N → X is said
to converge to x if for every open neighborhood U of x there is some NU such
that n ≥ NU implies that an ∈ U . If there is no x ∈ X for which a converges
to x, we say that a diverges.
Pn
Let a : N → R. We define s(a) : N → R by sn (a) = k=1 ak . We call sn (a)
the nth partial sum of the sequence a, and we call the sequence s(a) a series. If
there is some σ ∈ R such that s(a) converges to σ, we write
∞
X
ak = σ.
k=1
2
Goldbach
Euler [22, §110]: “If, as is commonly the case, we take the sum of a series to be
the aggregate of all of its terms, actually taken together, then there is no doubt
that only infinite series that converge continually closer to some value, the more
terms we actually add, can have sums”.
Euler Goldbach correspondence nos. 55, 161, 162.
3
Dirichlet
In 1837 Dirichlet proved that one can rearrange terms in an absolutely convergent series and not change the sum, and gave examples to show that this was
not the case for conditionally convergent series.
If a is a sequence and the series s(|a|) converges, we say that the series s(a)
is absolutely convergent. Because R is a complete metric space, a series being
absolutely convergent implies that it is convergent.
1
Dirichlet [14] and [15, p. 176, §101]
Elstrodt [20] and [19]
In the following theorem we prove that if a series converges absolutely, then
every rearrangement of it converges to the same value. Our proof follows Landau
[45, p. 157, Theorem 216].
Theorem 1. If a is a sequence for which s(a) converges absolutely and
∞
X
an = σ,
n=1
then for any bijection λ : N → N, the series s(a ◦ λ) converges to σ.
Proof. Let > 0, and let M be large enough so that
∞
X
|an | < .
n=M
Let r be large enough so that
{n : 1 ≤ n < M } ⊆ {λn : 1 ≤ n ≤ r}.
Fix m ≥ r, and let h : N → N be the sequence whose terms are the elements of
N \ {λn : 1 ≤ n ≤ m}
arranged in ascending order. If t + m ≥ max1≤n≤m λn then
{λn : 1 ≤ n ≤ m} ∪ {hn : 1 ≤ n ≤ t} = {n : 1 ≤ n ≤ t + m},
and hence
m
X
aλn +
n=1
t
X
ahn =
n=1
Taking t → ∞, we get
m
X
aλn +
n=1
t+m
X
an .
n=1
∞
X
ahn = σ;
n=1
the series s(a ◦ h) converges because for sufficiently large n, hn = n. Hence, for
every m ≥ r,
m
∞
∞
∞
X
X
X
X
aλn − σ = ahn ≤
|ahn | ≤
|an | < δ,
n=1
n=1
n=1
which shows that s(a ◦ λ) converges to σ.
2
n=M
4
Riemann rearrangement theorem
If a : N → R and λ : N → N is a bijection, we call the sequence a ◦ λ : N → R a
rearrangement of the sequence a.
Because N is a well-ordered set, if there are at least n elements in the set
{k ∈ N : ak ≥ 0} then it makes sense to talk about the nth nonnegative term in
the sequence a. If a were not a function from N to R but merely a function from
a countable set to R, it would not make sense to talk about the nth nonnegative
term in a or the nth negative term in a.
Riemann [60, pp. 96-97]
Our proof follows Landau [45, p. 158, Theorem 217].
Theorem 2 (Riemann rearrangement theorem). If a : N → R and s(a) converges but s(|a|) diverges, then for any nonnegative real number σ there is some
rearrangement b of a such that s(b) → σ.
Proof. Define p, q : N → R by
pn =
|an | + an
,
2
qn =
|an | − an
.
2
pn and qn are nonnegative, and satisfy pn − qn = an , pn + qn = |an |. If one of
s(p) or s(q) converges and the other diverges, we obtain a contradiction from
sn (a) =
n
X
k=1
ak =
n
X
(pk − qk ) =
k=1
n
X
pk −
k=1
n
X
qk = sn (p) − sn (q)
k=1
and the fact that s(a) converges. If both s(p) and s(q) converge, then we obtain
a contradiction from
sn (|a|) =
n
X
k=1
|ak | =
n
X
(pk + qk ) =
n
X
pk +
k=1
k=1
n
X
qk = sn (p) + sn (q)
k=1
and the fact that s(|a|) diverges. Therefore, both s(p) and s(q) diverge.
Because s(a) converges and s(|a|) diverges, there are infinitely many n with
an > 0 and there are infinitely many n with an < 0. Let Pn be the nth
nonnegative term in the sequence a, and let Qn be the absolute value of the nth
negative term in the sequence a. The fact that s(p) diverges implies that s(P )
diverges, and the fact that s(q) diverges implies that s(Q) diverges.
Let σ ≥ 0. We define sequences µ, ν : N → N by induction as follows. Let
µ1 be the least element of N such that
sµ1 (P ) > σ,
and with µ1 chosen, let ν1 be the least element of N such that
sµ1 (P ) − sν1 (Q) < σ.
3
Let m2 be the least element of N such that
sµ2 (P ) − sν1 (Q) > σ,
and with µ2 chosen, let ν2 be the least element of N such that
sµ2 (P ) − sν2 (Q) < σ.
It is straightforward to check that µ2 > µ1 and ν2 > ν1 .
Suppose that µ1 , . . . , µn and ν1 , . . . , νn have been chosen, that µn is the least
element of N such that
sµn (P ) − sνn−1 (Q) > σ,
that νn it the least element of N such that
sµn (P ) − sνn (Q) < σ
and that µn > µn−1 and νn > νn−1 . Let µn+1 be the least element of N such
that
sµn+1 (P ) − sνn (Q) > σ,
and with µn+1 chosen, let νn+1 be the least element of N such that
sµn+1 (P ) − sνn+1 (Q) < σ.
It is straightforward to check that µn+1 > µn and νn+1 > νn .
Define b : N → R by taking bn to be the nth term in
P1 , . . . , Pµ1 , −Q1 , . . . , −Qν1 , Pµ1 +1 , . . . , Pµ2 , −Qν1 +1 , . . . , −Qν2 , . . . ,
which, because the sequences µ and ν are strictly increasing, is a rearrangement
of the sequence a.
5
Symmetry
Don’t use order where it is accidental.
6
Nets
A directed set is a set D and a binary relation satisfying
• if m, n, p ∈ D, m n, and n p, then m p
• if m ∈ D, then m m
• if m, n ∈ D, then there is some p ∈ D such that m p and n p.
4
For example, let A be a set, let D be the set of all subsets of A, and say that
F G when F ⊆ G. Check that (D, ) is a directed set: for F, G ∈ D, we have
F ∪ G ∈ D, and F ∪ G is an upper bound for both F and G.
A net is a function from a directed set (D, ) to a set X. Let (X, τ ) be a
topological space, let S : (D, ) → (X, τ ) be a net, and let x ∈ X. We say that
S converges to x if for every U ∈ τ with x ∈ U there is some NU ∈ D such that
NU i implies that S(i) ∈ U . One proves that a topological space is Hausdorff
if and only if every net in this space converges to at most one point [41, p. 67,
Theorem 3].
A net S : (D, ) → R is said to be increasing if m n implies that S(m) ≤
S(n).
Lemma 3. If S : (D, ) → R is an increasing net and the range R of S has an
upper bound, then S converges to the supremum of R.
Proof. Because R is a subset of R that has an upper bound, it has a supremum,
call it σ. To say that σ is the supremum of R means that for all r ∈ R we have
r ≤ σ (σ is an upper bound) and that for all > 0 there is some r ∈ R with
σ − < r (nothing less than σ is an upper bound). Take > 0. There is some
r ∈ R with σ − < r . As r ∈ R, there is some n ∈ D with S(n ) = r . If
n n, then because S is increasing, S(n ) ≤ S(n), and hence
σ − < r = S(n ) ≤ S(n).
But S(n) ∈ R, so S(n) ≤ σ. Hence n n implies that |S(n) − σ| < , showing
that S converges to σ.
7
Unordered sums
Let A be a set, and let P0 (A) be the set of all finite subsets of A. Check that
(P0 (A), ⊆) is a directed set: if F, G ∈ P0 (A) then F ∪ G ∈ P0 (A) and F ∪ G
is an upper bound for both F and G. Let f : A → R be a function, and define
Sf : P0 (A) → R by
X
Sf (F ) =
f (a),
F ∈ P0 (A).
a∈F
If the net Sf converges, we say that the function f is summable, and we call the
element of R to which Sf converges the unordered sum of f , denoted by
X
f (a).
a∈A
If B is a subset of A, we say that f is summable over B if the restriction of f
to B is summable. If fB is the restriction of f to B and f is summable over B
(i.e. fB is summable), by
X
f (a)
a∈B
5
we mean
X
fB (a).
a∈B
Lemma 4. Suppose that f, g : A → R are functions and α, β ∈ R. If f and g
are summable, then αf + βg is summable and
X
X
X
(αf (a) + g(a)) = α
f (a) + β
g(a).
a∈A
a∈A
a∈A
P
P
Proof. Let σ1 = a∈A f (a) and σ2 = a∈A g(a), and set h = αf + βg. For
> 0, there is some F ∈ P0 (A) such that F ⊆ F ∈ P0 (A) implies that
|Sf (F ) − σ1 | < , and there is some G ∈ P0 (A) such that G ⊆ G ∈ P0 (A)
implies that |Sg (G) − σ2 | < . Let H = F ∪ G ∈ P0 (A). If H ⊆ H ∈ P0 (A),
then, as F ⊆ H and G ⊆ H,
X
(αf (a) + βg(a)) − ασ1 − βσ2 |Sh (H) − (ασ1 + βσ2 )| = a∈H
= |αSf (H) + βSg (H) − ασ1 − βσ2 |
≤
|α||Sf (H) − σ1 | + |β||Sg (H) − σ2 |
≤
|α| + |β|;
we write ≤ rather than < in the last inequality to cover the case where α = β =
0. It follows that Sh converges to ασ1 + βσ2 .
The following lemma is simple to prove and ought to be true, but should not
to be called obvious. For example, the Ces`aro sum of the sequence 1, −1, 1, −1, . . .
is 12 , while the Ces`
aro sum of the sequence 1, −1, 0, 1, −1, 0, . . . is 31 .
Lemma 5. If f : A → R is summable, then for any set C that contains A, the
function g : C → R defined by
(
f (c) c ∈ A
g(c) =
0
otherwise
is summable, and
X
f (a) =
a∈A
X
g(c).
c∈C
P
Proof. Let σ = a∈A f (a). For > 0, there is some F ∈ P0 (A) such that
F ⊆ F ∈ P0 (A) implies that |Sf (F ) − σ| < . If F ⊆ H ∈ P0 (C), then, as
6
F ⊆ H ∩ A ∈ P0 (A),
|Sg (H) − σ| =
=
=
X
g(c) − σ c∈H
X
X
g(c) +
g(c) − σ c∈H∩A
c∈H\A
X
X
f (a) +
0 − σ a∈H∩A
c∈H\A
=
|Sf (H ∩ A) − σ|
<
.
This shows that Sg converges to σ.
The previous two lemmas are useful, and also convince us that unordered
summation works similarly to finite sums. We now establish conditions under
which a function is summable.
Lemma 6. If f : A → R is nonnegative and there is some M ∈ R such that
F ∈ P0 (A) for all Sf (F ) ≤ M P
, then f is summable. If f : A → R is nonnegative
and summable, then Sf (F ) ≤ a∈A f (a) for all F ∈ P0 (A).
Proof. Suppose there is some M ∈ R such that if F ∈ P0 (A) then Sf (F ) ≤ M .
That is, M is an upper bound for the range of Sf . Because f is nonnegative,
the net Sf is increasing. We apply Lemma 3, which tells us that Sf converges
to the supremum of its range. That Sf converges
P means that f is summable.
Suppose that f is summable, and let σ = a∈A f (a). Suppose by contradiction that there is some F0 ∈ P0 (A) such that Sf (F0 ) > σ, and let = Sf (F0 )−σ.
Then there is some F ∈ P0 (A) such that F ⊆ F ∈ P0 (A) implies that
|Sf (F ) − σ| < . As F ⊆ F0 ∪ F ∈ P0 (A), we have |Sf (F0 ∪ F ) − σ| < , and
hence
Sf (F0 ∪ F ) < σ + = Sf (F0 ).
But F0 is contained in F0 ∪ F and f is nonnegative, so
Sf (F0 ) ≤ Sf (F0 ∪ F ),
which gives Sf (F0 ) < Sf (F0 ), a contradiction. Therefore, there is no F ∈ P0 (A)
for which Sf (F0 ) > σ.
Lemma 7. Suppose that f : A → R is a function and that A+ = {a ∈ A :
f (a) ≥ 0} and A− = {a ∈ A : f (a) ≤ 0}. Then, f is summable if and only if f
is summable over both A+ and A− . If f is summable, then
X
X
X
f (a) =
f (a) +
f (a).
a∈A
a∈A+
a∈A−
7
Proof. Suppose that f is summable. Because f is summable, there is some
E ∈ P0 (A) such that E ⊆ F ∈ P0 (A) implies that |Sf (F ) − σ| < 1. Define
E+ = {a ∈ E : f (a) ≥ 0} ∈ P0 (A+ ),
E− = {a ∈ E : f (a) ≤ 0} ∈ P0 (A− ).
If G ∈ P0 (A+ ) then E ⊆ G ∪ E ∈ P0 (A), and hence |Sf (G ∪ E) − σ| < 1. We
have
X
X
X
X
Sf+ (G) =
f (a) ≤
f (a) =
f (a) −
f (a),
a∈G
a∈G∪E+
a∈G∪E
a∈E−
and hence
Sf+ (G) ≤ Sf (G ∪ E) − Sf (E− ) < σ + 1 − Sf (E− ).
That is, σ + 1 − Sf (E− ) is an upper bound for the range of Sf+ . The net
Sf+ is increasing, hence applying Lemma 3 we get that Sf+ converges. That
is, f+ is summable. If H ∈ P0 (A− ), then E ⊆ H ∪ E ∈ P0 (A), and hence
|Sf (H ∪ E) − σ| < 1. We have
X
X
X
X
Sf− (H) =
f (a) ≥
f (a) =
f (a) −
f (a),
a∈H
a∈H∪E−
a∈H∪E
a∈E+
and then
Sf− (H) ≥ Sf (H ∪ E) − Sf (E+ ) > σ − 1 − Sf (E+ ),
showing that −σ + 1 + Sf (E+ ) is an upper bound for the net −Sf− . As −Sf−
is increasing, by Lemma 3 it converges, and it follows that Sf− converges. That
is, f− is summable.
Suppose that f is summable over both A+ and A− . Let f+ be the restriction
of f to A+ and let f+ be the restriction of f to A+ , and define g+ , g− : A → R
by
(
(
f (a) a ∈ A+
0
a ∈ A+
g+ (a) =
g− (a) =
0
a ∈ A− ,
f (a) a ∈ A− .
By Lemma 5, f+ being summable implies that g+ is summable, with
X
X
f+ (a) =
g+ (a),
a∈A+
a∈A
and f− being summable implies that g− is summable, with
X
X
f− (a) =
g− (a).
a∈A−
a∈A
But f = g+ + g− , so by Lemma 4 we get that f is summable, with
X
X
X
X
X
f (a) =
g+ (a) +
g− (a) =
f+ (a) +
f− (a).
a∈A
a∈A
a∈A
a∈A+
8
a∈A−
If f : A → R is a function, we define |f | : A → R by |f |(a) = |f (a)|.
Theorem 8. If f : A → R is a function, then f is summable if and only if |f |
is summable.
Proof. Let A+ = {a ∈ A : f (a) ≥ 0} and A− = {a ∈ A : f (a) ≤ 0}, and let f+
and f− be the restrictions of f to A+ and A− respectively. Suppose that f is
summable. Then by Lemma 7 we get that f+ is summable and f− is summable.
Let F ∈ P0 (A) and write F+ = {a ∈ F : f (a) ≥ 0}, F− = {a ∈ F : f (a) ≤ 0}.
We have
X
X
X
S|f | (F ) =
|f (a)| =
f (a) −
f (a) = Sf+ (F+ ) − Sf− (F− ).
a∈F
a∈F+
a∈F−
P
But by Lemma 6, because the net Sf+ is increasing we have Sf+ (F+ ) ≤ a∈A+ f+ (a),
P
and because the net −Sf− is increasing we have −Sf− (F− ) ≤ − a∈A− f− (a).
P
P
Therefore, a∈A+ f+ (a)− a∈A− f− (a) is an upper bound for the range of S|f | .
Moreover, S|f | is increasing, so by Lemma 6 it follows that S|f | converges, i.e.
that |f | is summable.
Suppose that |f | is summable. By Lemma 6, for any F ∈ P0 (A+ ) we have
X
Sf+ (F ) = S|f | (F ) ≤
|f |(a),
a∈A
P
i.e., a∈A |f |(a) is an upper bound for the range of Sf+ . As Sf+ is increasing,
by Lemma 6 it follows that Sf+ converges, i.e., that f+ is summable. Because
−Sf− is increasing, we likewise get that −Sf− converges and hence that Sf−
converges, i.e. that f− is summable. Now applying Lemma 7, we get that f is
summable.
Theorem 9. If f : A → R is summable, then {a ∈ A : f (a) 6= 0} is countable.
Proof. Suppose by contradiction that {a ∈ A : f (a) 6= 0} is uncountable. We
have
[
1
{a ∈ A : f (a) 6= 0} = {a ∈ A : |f (a)| > 0} =
a ∈ A : |f (a)| ≥
.
n
n∈N
Since this is a countable union, there is some n ∈ N such that a ∈ A : |f (a)| ≥ n1
is uncountable; in particular, this set is infinite. Because f is summable, by Theorem 8 we have that |f | is summable, with unordered sum σ. Hence, there is
some F1 ∈ P0 (A) such that
F1 ⊆ F ∈ P0 (A)
implies that |S|f | (F ) − σ| < 1.
Let F be a finite subset of a ∈ A : |f (a)| ≥ n1 with at least n(σ + 1) elements.
Then
X
X
1
S|f | (F ∪ F1 ) =
|f (a)| ≥
|f (a)| ≥ n(σ + 1) · = σ + 1.
n
a∈F ∪F1
a∈F
But F1 ⊆ F ∪ F1 ∈ P0 (A), so S|f | (F ∪ F1 ) < σ + 1, a contradiction. Therefore,
{a ∈ A : f (a) 6= 0} is countable.
9
8
References
McArthur [52]
Schaefer [64, p. 120]
Roytvarf [63, p. 282]
McShane [53]
Diestel [13, Chapter VI]
Remmert [59, p. 29]
Sorenson [67]
Lattice sums [18]
Kadets and Kadets [40]
Manning [50]
Bottazini [2]
Boyer [4]
Weil [74]
Smithies [66]
Dugac [16]
Grattan-Guinness [31] and [33] and [32] and [30]
Whiteside [75]
Schaefer [65]
Cauchy [5]
Polya [57]
Lakatos [44]
Krantz [43]
Cunha [12]
Youschkevitch [77]
Tweddle [70] and [71]
Bromwich [6, p. 74, Art. 28]
Laugwitz [46], [47], [48]
Tucciarone [69]
Fraser [27]
Cowen [11]
Spence [68]
Jahnke [38]
Epple [21]
Mascr´e [51]
Rosenthal [61]
Freniche [28]
Goursat [29, p. 348]
9
Probability
Baker [1]
Nathan [55]
Nover and Harris [56]
10
Colyvan [10]
Liouville [49, pp. 74–75]
Chrystal [9, p. 118]
Jordan [39, p. 277, Theorem 291]
Cayley [8, a]
Harkness and Morley [35, p. 66]
Hofmann [37]
Ferreir´
os [26]
Ferraro [25] and [23] and [24]
Brouncker [7]
Roy [62]
Wallis [72]
Bourbaki [3, p. 261, chapter III, §5.1]
Pringsheim [58]
Dutka [17]
Gr¨
unbaum [34]
Hinton and Martin [36]
Gersonides [42]
Watling [73]
Moore [54]
Wojtaszczyk [76, Chapter 7]
References
[1] Alan Baker, Putting expectations in order, Philosophy of Science 74 (2007),
no. 5, 692–700.
[2] Umberto Bottazini, The higher calculus: A history of real and complex
analysis from Euler to Weierstrass, Springer, 1986, Translated from the
Italian by Warren Van Egmond.
[3] Nicolas Bourbaki, General topology, chapters 1–4, Elements of Mathematics, Springer, 1989.
[4] Carl B. Boyer, The history of the calculus and its conceptual development,
Dover Publications, New York, 1959.
[5] Robert E. Bradley and C. Edward Sandifer, Cauchy’s Cours d’analyse: an
annotated translation, Sources and Studies in the History of Mathematical
and Physical Sciences, Springer, 2009.
[6] T. J. I’A. Bromwich, An introduction to the theory of infinite series, second
ed., Macmillan and Co., London, 1931.
[7] William Brouncker, The squaring of the hyperbola, by an infinite series of
rational numbers, together with its demonstration, by that eminent mathematician, the right honourable the Lord Viscount Brouncker, Phil. Trans.
3 (1668), 645–649.
11
[8] Arthur Cayley, Series, Encyclopædia Britannica, vol. XXI, Charles Scribner’s Sons, New York, ninth ed., 1886, Collected Mathematical Papers, vol.
XI, pp. 617–627, pp. 677–682.
[9] George Chrystal, Algebra: An elementary text-book for the higher classes
of secondary schools and for colleges, part II, Adam and Charles Black,
Edinburgh, 1889.
[10] Mark Colyvan, No expectations, Mind 115 (2006), no. 459, 695–702.
[11] C. C. Cowen, K. R. Davidson, and R. P. Kaufman, Rearranging the alternating harmonic series, Amer. Math. Monthly 87 (1980), no. 10, 817–819.
[12] A. J. Franco de Oliveira, Anast´
acio da Cunha and the concept of convergent
series, Arch. Hist. Exact Sci. 39 (1988), no. 1, 1–12.
[13] Joseph Diestel, Sequences and series in Banach spaces, Graduate Texts in
Mathematics, vol. 92, Springer, 1984.
[14] Gustav Lejeune Dirichlet, Beweis des Satzes, dass jede unbegrenzte arithmetische Progression, deren erstes Glied und Differenz ganze Zahlen ohne
gemeinschaftlichen Factor sind, unendlich viele Primzahlen enth¨
alt, Abhandlungen der K¨
oniglichen Akademie der Wissenschaften zu Berlin (1837),
45–81, Werke, vol. I, pp. 313–342.
[15] P. G. L. Dirichlet, Lectures on number theory, History of Mathematics,
vol. 16, American Mathematical Society, Providence, RI, 1999, Supplements by R. Dedekind, translated from the German by John Stilwell.
´ ements d’analyse de Karl Weierstrass, Arch. History Exact
[16] Pierre Dugac, El´
Sci. 10 (1973), no. 1/2, 41–176.
[17] Jacques Dutka, On the St. Petersburg paradox, Arch. Hist. Exact Sci. 39
(1988), no. 1, 13–39.
[18] Ulrich Eckhardt, Gittersummen der Festk¨
orperphysik. Wie sinnvoll ist die
Sprache des Unendlichen f¨
ur die Dinge des allt¨
aglichen Lebens?, Mitt.
Math. Ges. Hamburg 29 (2010), 71–96.
[19] J¨
urgen Elstrodt, Annotationes historicae, Mitt. Dtsch. Math.-Ver. (1998),
no. 1, 23–24.
[20] J¨
urgen Elstrodt, The life and work of Gustav Lejeune Dirichlet (1805–
1859), Analytic Number Theory: A Tribute to Gauss and Dirichlet
(William Duke and Yuri Tschinkel, eds.), Clay Mathematics Proceedings,
vol. 7, American Mathematical Society, Providence, RI, 2005, pp. 1–38.
[21] Moritz Epple, The end of the science of quantity: foundations of analysis, 1860–1910, A History of Analysis (Hans Niels Jahnke, ed.), History
of Mathematics, vol. 24, American Mathematical Society, Providence, RI,
2003, pp. 291–324.
12
[22] Leonhard Euler, Institutiones calculi differentialis, vol. I, Academia Scientiarum Imperialis Petropolitanae, St. Petersburg, 1755, E212, Opera omnia
I.10.
[23] Giovanni Ferraro, The first modern definition of the sum of a divergent
series: an aspect of the rise of 20th century mathematics, Arch. Hist. Exact
Sci. 54 (1999), no. 2, 101–135.
[24]
, True and fictitious quantities in Leibniz’s theory of series, Studia
Leibnitiana 32 (2000), no. 1, 43–67.
[25]
, The rise and development of the theory of series up to the early
1820s, Sources and Studies in the History of Mathematics and Physical
Sciences, Springer, 2008.
[26] Jos´e Ferreir´
os, Labyrinth of thought: A history of set theory and its role in
modern mathematics, second ed., Birkh¨auser, 2007.
[27] Craig G. Fraser, The calculus as algebraic analysis: some observations on
mathematical analysis in the 18th century, Arch. Hist. Exact Sci. 39 (1989),
no. 4, 317–335.
[28] Francisco J. Freniche, On Riemann’s rearrangement theorem for the alternating harmonic series, Amer. Math. Monthly 117 (2010), no. 5, 442–448.
´
[29] Edouard
Goursat, A course in mathematical analysis, vol. I, Ginn and
Company, Boston, 1904, Translated from the French by Earle Raymond
Hedrick.
[30] Ivor Grattan-Guinness, Bolzano, Cauchy and the “new analysis” of the
early nineteenth century, Arch. Hist. Exact Sci. 6 (1970), no. 5, 372–400.
[31]
, The development of the foundations of mathematical analysis from
Euler to Riemann, MIT Press, 1970.
[32]
, The emergence of mathematical analysis and its foundational
progress, 1780–1880, From the Calculus to Set Theory, 1630–1910 (Ivor
Grattan-Guinness, ed.), Princeton University Press, 1980, pp. 94–148.
[33]
, Thus it mysteriously appears: impressions of Laplace’s use of series, Rechnen mit dem Unendlichen: Beitr¨age zur Entwicklung eines kontroversen Gegenstandes (Detlef D. Spalt, ed.), Birkh¨auser, 1990, pp. 95–
102.
[34] Adolf Gr¨
unbaum, Can an infinitude of operations be performed in a finite
time?, Br. J. Philos. Sci. 20 (1969), no. 3, 203–218.
[35] James Harkness and Frank Morley, A treatise on the theory of functions,
Macmillan, London, 1893.
13
[36] J. M. Hinton and C. B. Martin, Achilles and the tortoise, Analysis 14
(1954), no. 3, 56–68.
[37] Joseph Ehrenfried Hofmann, On the discovery of the logarithmic series and
its development in England up to Cotes, National Mathematics Magazine
14 (1939), no. 1, 37–45.
[38] Hans Niels Jahnke, Algebraic analysis in the 18th century, A History of
Analysis (Hans Niels Jahnke, ed.), History of Mathematics, vol. 24, American Mathematical Society, Providence, RI, 2003, pp. 105–136.
[39] M. C. Jordan, Cours d’analyse de l’´ecole Polytechnique, tome premier, second ed., Gauthier-Villars et fils, Paris, 1893.
[40] Mikhail I. Kadets and Vladimir M. Kadets, Series in Banach spaces: Conditional and unconditional convergence, Operator Theory: Advances and
Applications, vol. 94, Birkh¨auser, 1997, Translated from the Russian by
Andrei Iacob.
[41] John L. Kelley, General topology, University Series in Higher Mathematics,
D. Van Nostrand Company, 1955.
[42] George Kohler, Medieval infinities in mathematics and the contribution of
Gersonides, History of Philosophy Quarterly 23 (2006), no. 2, 95–116.
[43] Steven G. Krantz and Jeffery D. McNeal, Creating more convergent series,
Amer. Math. Monthly 111 (2004), no. 1, 32–38.
[44] Imre Lakatos, Proofs and refutations, Cambridge University Press, 1976.
[45] Edmund Landau, Differential and integral calculus, Chelsea Publishing
Company, New York, 1951, Translated from the German by Melvin Hausner and Martin Davis.
[46] Detlef Laugwitz, Definite values of infinite sums: Aspects of the foundations
of infinitesimal analysis around 1820, Arch. Hist. Exact Sci. 39 (1989),
no. 3, 195–245.
[47]
, Controversies about numbers and functions, The Growth of Mathematical Knowledge (Emily Grosholz and Herbert Breger, eds.), Sythese
Library, vol. 289, Kluwer, 2000, pp. 177–198.
[48]
, Bernhard Riemann 1826–1866: Turning points in the conception
of mathematics, Birkh¨
auser, 2008, Translated from the German by Abe
Shenitzer.
[49] Jesper L¨
utzen, Joseph Liouville 1809–1882: master of pure and applied
mathematics, Studies in the History of Mathematics and Physical Sciences,
vol. 15, Springer, 1990.
14
[50] Kenneth R. Manning, The emergence of the Weierstrassian approach to
complex analysis, Arch. Hist. Exact Sci. 14 (1975), no. 4, 297–383.
[51] David Mascr´e, 1867 Bernhard Riemann, posthumous thesis on the representation of functions by trigonometric series, Landmark Writings in Western Mathematics 1640–1940 (Ivor Grattan-Guinness, ed.), Elsevier, 2005,
pp. 491–505.
[52] C. W. McArthur, Series with sums invariant under rearrangement, Amer.
Math. Monthly 75 (1968), 729–731.
[53] E. J. McShane, Partial orderings and Moore-Smith limits, Amer. Math.
Monthly 59 (1952), no. 1, 1–11.
[54] C. N. Moore, Summability of series, Amer. Math. Monthly 39 (1932), no. 2,
62–71.
[55] Amos Nathan, False expectations, Philosophy of Science 51 (1984), no. 1,
128–136.
[56] Harris Nover and Alan H´ajek, Vexing expectations, Mind 113 (2004),
no. 450, 237–249.
[57] George P´
olya and G´
abor Szeg˝o, Problems and theorems in analysis, volume
I, Die Grundlehren der mathematischen Wissenschaften, vol. 193, Springer,
1972, Translated from the German by D. Aeppli.
[58] Alfred Pringsheim, Irrationalzahlen und Konvergenz unendlicher Prozesse,
Enzyklop¨
adie der Mathematischen Wissenschaften mit Einschluss ihrer Anwendungen. Band I, 1. Teil (Wilhelm Franz Meyer, ed.), B. G. Teubner,
Leipzig, 1898–1904, pp. 47–146.
[59] Reinhold Remmert, Theory of complex functions, Graduate Texts in Mathematics, vol. 122, Springer, 1991, Translated from the German by Robert
B. Burckel.
[60] Bernhard Riemann, Ueber die Darstellbarkeit einer Function durch eine
trigonometrische Reihe, Abhandlungen der K¨oniglichen Gesellschaft der
Wissenschaften zu G¨
ottingen 13 (1866/1867), 87–132, Gesammelte mathematische Werke, second ed., vol. II, 227–271.
[61] Peter Rosenthal, The remarkable theorem of L´evy and Steinitz, Amer.
Math. Monthly 94 (1987), no. 4, 342–351.
[62] Ranjan Roy, Sources in the development of mathematics, Cambridge University Press, 2011, Infinite series and products from the fifteenth to the
twenty-first century.
[63] Alexander A. Roytvarf, Thinking in problems: How mathematicians find
creative solutions, Birkh¨auser, 2013.
15
[64] H. H. Schaefer, Topological vector spaces, second ed., Graduate Texts in
Mathematics, vol. 3, Springer, 1999, With the assistance of M. P. Wolff.
[65] Paul Schaefer, Sums of rearranged series, College Math. J. 17 (1986), no. 1,
66–70.
[66] F. Smithies, Cauchy’s conception of rigour in analysis, Arch. Hist. Exact
Sci. 36 (1986), no. 1, 41–61.
[67] Henrik Kragh Sørensen, Throwing some light on the vast darkness that is
analysis: Niels Henrik Abel’s critical revision and the concept of absolute
convergence, Centaurus 52 (2010), no. 1, 38–72.
[68] Lawrence E. Spence, How many elements are in a union of sets?, The
Mathematics Teacher 80 (1987), no. 8, 666–670, 681.
[69] John Tucciarone, The development of the theory of summable divergent
series from 1880 to 1925, Arch. Hist. Exact Sci. 10 (1973), no. 1/2, 1–40.
[70] Ian Tweddle, James Stirling’s early work on acceleration of convergence,
Arch. Hist. Exact Sci. 45 (1992), no. 2, 105–125.
[71]
, James Stirling’s methodus differentialis: An annotated translation
of Stirling’s text, Sources and Studies in the History of Mathematics and
Physical Sciences, Springer, 2003.
[72] John Wallis, The arithmetic of infinitesimals, Sources and Studies in the
History of Mathematics and Physical Sciences, Springer, 2004, Translated
from the Latin and with an introduction by Jaequeline A. Stedall.
[73] J. Watling, The sum of an infinite series, Analysis 13 (1952), no. 2, 39–46.
[74] Andr´e Weil, Elliptic functions according to Eisenstein and Kronecker,
Ergebnisse der Mathematik und ihrer Grenzgebiete, vol. 88, Springer, 1976.
[75] Derek Thomas Whiteside, Patterns of mathematical thought in the later
seventeenth century, Arch. Hist. Exact Sci. 1 (1961), no. 3, 179–388.
[76] P. Wojtaszczyk, A mathematical introduction to wavelets, London Mathematical Society Student Texts, vol. 37, Cambridge University Press, 1997.
[77] A. P. Youschkevitch, The concept of function up to the middle of the 19th
century, Arch. Hist. Exact Sci. 16 (1976), no. 1, 37–85.
16